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Harmonics. October 29, 2012. Where Were We?. We’re halfway through grading the mid-terms. For the next two weeks: more acoustics It’s going to get worse before it gets better Note: I’ve posted a supplementary reading on today’s lecture to the course web page. What have we learned?

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harmonics
Harmonics

October 29, 2012

where were we
Where Were We?
  • We’re halfway through grading the mid-terms.
  • For the next two weeks: more acoustics
    • It’s going to get worse before it gets better
  • Note: I’ve posted a supplementary reading on today’s lecture to the course web page.
  • What have we learned?
    • How to calculate the frequency of a periodic wave
    • How to calculate the amplitude, RMS amplitude, and intensity of a complex wave
  • Today: we’ll start putting frequency and intensity together
complex waves
Complex Waves
  • When more than one sinewave gets combined, they form a complex wave.
  • At any given time, each wave will have some amplitude value.
    • A1(t1) := Amplitude value of sinewave 1 at time 1
    • A2(t1) := Amplitude value of sinewave 2 at time 1
  • The amplitude value of the complex wave is the sum of these values.
    • Ac(t1) = A1 (t1) + A2 (t1)
complex wave example
Complex Wave Example
  • Take waveform 1:
    • high amplitude
    • low frequency

+

  • Add waveform 2:
    • low amplitude
    • high frequency

=

  • The sum is this complex waveform:
greatest common denominator
Greatest Common Denominator
  • Combining sinewaves results in a complex periodic wave.
  • This complex wave has a frequency which is the greatest common denominator of the frequencies of the component waves.
  • Greatest common denominator = biggest number by which you can divide both frequencies and still come up with a whole number (integer).
  • Example:
    • Component Wave 1: 300 Hz
    • Component Wave 2: 500 Hz
    • Fundamental Frequency: 100 Hz
slide6
Why?
  • Think: smallest common multiple of the periods of the component waves.
  • Both component waves are periodic
    • i.e., they repeat themselves in time.
  • The pattern formed by combining these component waves...
    • will only start repeating itself when both waves start repeating themselves at the same time.
  • Example: 3 Hz sinewave + 5 Hz sinewave
for example
For Example
  • Starting from 0 seconds:
    • A 3 Hz wave will repeat itself at .33 seconds, .66 seconds, 1 second, etc.
    • A 5 Hz wave will repeat itself at .2 seconds, .4 seconds, .6 seconds, .8 seconds, 1 second, etc.
  • Again: the pattern formed by combining these component waves...
    • will only start repeating itself when they both start repeating themselves at the same time.
    • i.e., at 1 second
slide8

3 Hz

.33 sec.

.66

1.00

5 Hz

.20

1.00

.40

.60

.80

slide9

1.00

Combination of 3 and 5 Hz waves

(period = 1 second)

(frequency = 1 Hz)

tidbits
Tidbits
  • Important point:
    • Each component wave will complete a whole number of periods within each period of the complex wave.
  • Comprehension question:
    • If we combine a 6 Hz wave with an 8 Hz wave...
    • What should the frequency of the resulting complex wave be?
  • To ask it another way:
    • What would the period of the complex wave be?
slide11

6 Hz

.17 sec.

.33

.50

8 Hz

.125

.25

.375

.50

slide12

.50 sec.

1.00

Combination of 6 and 8 Hz waves

(period = .5 seconds)

(frequency = 2 Hz)

fourier s theorem
Fourier’s Theorem
  • Joseph Fourier (1768-1830)
    • French mathematician
    • Studied heat and periodic motion
  • His idea:
    • any complex periodic wave can be constructed out of a combination of different sinewaves.
  • The sinusoidal (sinewave) components of a complex periodic wave = harmonics
the dark side
The Dark Side
  • Fourier’s theorem implies:
    • sound may be split up into component frequencies...
    • just like a prism splits light up into its component frequencies
spectra
Spectra
  • One way to represent complex waves is with waveforms:
    • y-axis: air pressure
    • x-axis: time
  • Another way to represent a complex wave is with a power spectrum (or spectrum, for short).
  • Remember, each sinewave has two parameters:
    • amplitude
    • frequency
  • A power spectrum shows:
    • intensity (based on amplitude) on the y-axis
    • frequency on the x-axis
two perspectives
Two Perspectives

WaveformPower Spectrum

+

+

=

=

harmonics

example
Example
  • Go to Praat
  • Generate a complex wave with 300 Hz and 500 Hz components.
  • Look at waveform and spectral views.
  • And so on and so forth.
fourier s theorem part 2
Fourier’s Theorem, part 2
  • The component sinusoids (harmonics) of any complex periodic wave:
    • all have a frequency that is an integer multiple of the fundamental frequency of the complex wave.
  • This is equivalent to saying:
    • all component waves complete an integer number of periods within each period of the complex wave.
example1
Example
  • Take a complex wave with a fundamental frequency of 100 Hz.
  • Harmonic 1 = 100 Hz
  • Harmonic 2 = 200 Hz
  • Harmonic 3 = 300 Hz
  • Harmonic 4 = 400 Hz
  • Harmonic 5 = 500 Hz
  • etc.
deep thought time
Deep Thought Time
  • What are the harmonics of a complex wave with a fundamental frequency of 440 Hz?
  • Harmonic 1: 440 Hz
  • Harmonic 2: 880 Hz
  • Harmonic 3: 1320 Hz
  • Harmonic 4: 1760 Hz
  • Harmonic 5: 2200 Hz
  • etc.
  • For complex waves, the frequencies of the harmonics will always depend on the fundamental frequency of the wave.
a new spectrum
A new spectrum
  • Sawtooth wave at 440 Hz
  • Also check out a sawtooth wave at 150 Hz
another representation
Another Representation
  • Spectrograms
    • = a three-dimensional view of sound
  • Incorporate the same dimensions that are found in spectra:
    • Intensity (on the z-axis)
    • Frequency (on the y-axis)
  • And add another:
    • Time (on the x-axis)
  • Back to Praat, to generate a complex tone with component frequencies of 300 Hz, 2100 Hz, and 3300 Hz
something like speech
Something Like Speech
  • Check this out:
  • One of the characteristic features of speech sounds is that they exhibit spectral change over time.

source: http://www.haskins.yale.edu/featured/sws/swssentences/sentences.html

harmonics and speech
Harmonics and Speech
  • Remember: trilling of the vocal folds creates a complex wave, with a fundamental frequency.
  • This complex wave consists of a series of harmonics.
  • The spacing between the harmonics--in frequency--depends on the rate at which the vocal folds are vibrating (F0).
  • We can’t change the frequencies of the harmonics independently of each other.
  • Q: How do we get the spectral changes we need for speech?
resonance
Resonance
  • Answer: we change the intensity of the harmonics
  • Listen to this:
  • source: http://www.let.uu.nl/~audiufon/data/e_boventoon.html
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